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   -> Volume 7, Issue 1

Preprint: Biorthogonal Multiwavelets on the Interval
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"Angela Kunoth" (

PostPosted: Sun Jan 11, 1998 4:00 pm    
Subject: Preprint: Biorthogonal Multiwavelets on the Interval
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#7 Preprint: Biorthogonal Multiwavelets on the Interval

Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

Wolfgang Dahmen, Bin Han, Rong-Qing Jia and Angela Kunoth


Starting with Hermite cubic splines as primal multigenerator, first a
dual multigenerator on the whole real line is constructed which
consists of continuous functions, has small support and is exact of
order two. We then derive multiresolution sequences on the interval
while retaining the polynomial exactness on the primal and dual side.
This guarantees moment conditions of the corresponding wavelets. The
concept of stable completions by Carnicer, Dahmen, Pena is then used
to construct corresponding primal and dual multiwavelets on the
interval as follows. An appropriate variation of what is known as
hierarchical basis in finite element methods is shown to be an initial
completion. This is then in a second step projected into the desired
complements spanned by compactly supported biorthogonal multiwavelets.
The masks of all multigenerators and multiwavelets are finite so that
decomposition and reconstruction algorithms are simple and efficient.
Furthermore, in addition to Jackson estimates which follow from the
exactness, one can also show Bernstein inequalities for the primal and
dual multiresolution. Consequently, sequence norms for the
coefficients based on such multiwavelet expansions characterize
Sobolev norms of Sobolev spaces H^s([0,1]) for s between -0.824926 and
2.5. In particular, the multiwavelets form Riesz bases for L_2 on the
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